- #1
epkid08 said:Am I missing something, why aren't you using
[tex]\zeta(s) = \frac{1}{1-2^{1-s}}\sum_{n\geq1}\frac{(-1)^{n-1}}{n^{s}}[/tex]
as your definition of zeta?
The definition of zeta in your proof is invalid for [tex] \text{Re} < 1[/tex].
amitjohar said:Disproof means showing a counter example, where the real part of the non-trivial zero is not half. If there is no counter example, then it cannot be accepted as disproof.
choe said:hmm...
amitjohar said:Well not for Riemann Hypothesis. What I meant to say is that in the case of RH, the only way to disprove is to provide a counter-example. All the disproofs without any counter-example will never be right in the case of RH. All such RH disproofs, attempted by numerous people were scrapped. Therefore it is reasonable to conclude that if a disproof is shown in RH, it is only going to be through counter-example. Otherwise it simply cannot work. Now in other cases, disproof without counter example can work. But definitely not for RH
amitjohar said:I agree with you but the main problem lies that RH may not really be false. If it is true, then there is no point in trying to create a disproof. There is no reason to think that the RH is wrong. Most mathematicians think it is true.
RH maybe unprovable because of Godel's theorem. Godel's theorem states that for any given mathematical system, there are mathematical truths that cannot be proven within that system. The Riemann Hypothesis maybe one of these. If Godel theorem holds, then RH can never be proven true and it can also never be proven false.
amitjohar said:I agree with you but the main problem lies that RH may not really be false. If it is true, then there is no point in trying to create a disproof. There is no reason to think that the RH is wrong. Most mathematicians think it is true. RH maybe unprovable because of Godel's theorem. Godel's theorem states that for any given mathematical system, there are mathematical truths that cannot be proven within that system. The Riemann Hypothesis maybe one of these. If Godel theorem holds, then RH can never be proven true and it can also never be proven false.
atomthick said:That's interesting because if you prove that RH is not provable due to Godel's theorem that means RH is actually true because you can prove it to be false by searching for a root not on the critical line (of course there are an infinite number of them but nonetheless there is a way to prove it to be false).
What terrible "logic"! Your argument is that such proofs have not worked, so it is "reasonable to conclude" that they can't work, which then becomes, it is impossible!amitjohar said:Well not for Riemann Hypothesis. What I meant to say is that in the case of RH, the only way to disprove is to provide a counter-example. All the disproofs without any counter-example will never be right in the case of RH. All such RH disproofs, attempted by numerous people were scrapped. Therefore it is reasonable to conclude that if a disproof is shown in RH, it is only going to be through counter-example. Otherwise it simply cannot work. Now in other cases, disproof without counter example can work. But definitely not for RH
This is not how proof by induction works, but you probably already knew that. I liked your post though, especially where you put ALL in caps.zhentil said:I think that since ALL disproofs have been scrapped, RH is true by induction.
lostcauses10x said:been a bit bored so...
The first trivial zero is at -2, the pattern from -2 is every even negative intiger continuing from -2.
So -2 - (-2)= 0
The value of the Reiman zeta function at 0 = -1/2
The offset this value designates is due to the pole at the integer of 1.
1+(-1/2)=1/2
This is the line the non trivial zeros will occur at.
The Riemann hypothesis is a mathematical conjecture proposed by Bernhard Riemann in 1859. It states that all non-trivial zeros of the Riemann zeta function lie on the critical line with real part 1/2.
The Riemann hypothesis has important implications in number theory, specifically in the distribution of prime numbers. It also has connections to other areas of mathematics, such as complex analysis and algebraic geometry.
The renewed disproof of the Riemann hypothesis is a recent attempt to disprove the conjecture using a new mathematical approach. It has gained attention in the mathematical community, but has not been widely accepted as a valid disproof.
No, the Riemann hypothesis has not been proven or disproven. It remains one of the most famous unsolved problems in mathematics, and has been a subject of much research and debate among mathematicians.
If the Riemann hypothesis is disproven, it would have significant implications in the study of prime numbers and potentially lead to new developments in number theory. It would also challenge many current theories and open up new avenues for research in mathematics.